Question: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-2x-6y &= 9 \\ -x+2y &= 1\end{align*}$
Answer: Begin by moving the $x$ -term in the second equation to the right side of the equation. $2y = x+1$ Divide both sides by $2$ to isolate $y$ $y = {\dfrac{1}{2}x + \dfrac{1}{2}}$ Substitute this expression for $y$ in the first equation. $-2x-6({\dfrac{1}{2}x + \dfrac{1}{2}}) = 9$ $-2x - 3x - 3 = 9$ Simplify by combining terms, then solve for $x$ $-5x - 3 = 9$ $-5x = 12$ $x = -\dfrac{12}{5}$ Substitute $-\dfrac{12}{5}$ for $x$ back into the top equation. $-2( -\dfrac{12}{5})-6y = 9$ $\dfrac{24}{5}-6y = 9$ $-6y = \dfrac{21}{5}$ $y = -\dfrac{7}{10}$ The solution is $\enspace x = -\dfrac{12}{5}, \enspace y = -\dfrac{7}{10}$.